1. Field of the Invention
This invention relates to the field of vacuum chambers. More specifically, the invention comprises a novel construction for a vacuum chamber in which the wall members are subjected primarily to tensile forces. Applications for the vacuum chamber are also disclosed.
2. Description of the Related Art
Vacuum chambers have been in existence for many years. They generally have thick and strong walls, so that they can resist the compressive forces exerted by the surrounding atmosphere. The walls must resist simple compressive forces (hoop and axial forces), and must generally be thick enough to prevent buckling. Those skilled in the art will know that buckling instability typically defines the failure limit of a vacuum chamber.
Traditional designs have been quite heavy. For earth-bound vacuum chambers, weight is frequently not a concern. However, for mobile chambers, weight can be a very significant concern. The weight associated with traditional vacuum chambers can be highly significant in such an environment.
A very light-weight vacuum chamber raises the possibility of displacing a greater weight of atmosphere than the weight of the vacuum chamber itself. Such a design could achieve positive buoyance, creating a rigid “vacuum balloon.” There are numerous applications for such a device, some of which will be disclosed subsequently. The reader will appreciate at this point, however, that a very light-weight vacuum chamber has significant advantages over a traditional vacuum chamber.
The walls of traditional vacuum chambers have experienced primarily compressive forces. Such a chamber typically fails well before the ultimate compressive limit of the material used, since the chamber fails by buckling. Compressive forces tend to produce an inherently unstable situation. The walls must be thick enough to provide anti-buckling stability.
In determining the chamber's required strength, two failure modes must be considered—yield failure and deformation (buckle) failure. The mathematics of yield failure is simpler and will considered first.
A simple analysis for a conventional thin-walled vacuum chamber (having a homogenous wall, unlike the present invention) is helpful. By taking a section that cuts the cylinder into two semi-cylinders, the minimum required wall thickness (t) (neglecting safety factors) to overcome yield failure can be computed using the cylinder's radius (R), the yield strength of the material used (Y), and the external pressure (P). In order to prevent yield failure, the following must be true:tY>RP
Given that the thickness to radius ratio is size invariant, one can simplify the expression by setting the radius equal to unity, and thus consider all other linear dimensions to be fractions of the chamber's radius. Thus, a simpler expression for the required wall thickness becomes:
                    t        >                  P          Y                                    (                  Equation          ⁢                                          ⁢          1                )            
In order to achieve positive buoyancy, the mass of the chamber must be less than the mass of the air displaced. If ρ is the density of the chamber material, then the following equation must be satisfied:2πRtρ<πR2ρair 
If the radius is again set to unity, and the thickness expressed as a fraction thereof, then a simplified expression for the maximum wall thickness allowable while still achieving buoyancy can be written as:
                    t        <                              1            2                    ⁢                                    ρ              air                        ρ                                              (                  Equation          ⁢                                          ⁢          2                )            
Substituting Equation 2 into Equation 1 gives the following expression:
            Y      ρ        ·                  ρ        air            P        >  2
The left term on the left side of this equation is a function of the material selected. The right term on the left side of this equation depends on the properties of the air. The reader should note that this term is temperature dependent but not pressure dependent (at a fixed temperature). One can compute a “critical temperature” for each material indicating the temperature above which there is no theoretical hope of constructing a positively buoyant vacuum chamber (since the air density will be so low above that temperature that the mass displaced by the vacuum chamber will be lighter than the chamber itself). The following table presents the strength to density ratios and critical temperatures for several available materials, in units of air pressure and density, respectively:
TABLE IYieldDensityratioTcriticalMaterialstrength (ksi)(g/cm3)(@15° C.)(° C.)2014-T6 Aluminum602.791.79−15304 Stainless Steel808.030.82−215Am 1004-T61 Magnesium221.831.00−129Ti—6Al—4V Titanium1344.432.5290Kevlar 49 FRP471.452.68113Carbon fiber composite2501.6212.861580
Thus, if one only analyses the yield failure, then one must simply choose a material from TABLE I, make sure that the ambient temperature is below the critical temperature listed, and then choose an appropriate wall thickness based on the prior equations.
As an example, the following thicknesses would be appropriate for a chamber having a radius of one meter at standard sea-level conditions:
TABLE IIMaterialThickness (mm)Ti-6Al-4V Titanium0.11–0.14Kevlar 49 FRP0.31–0.42Carbon fiber composite0.06–0.38
Of course, one must also consider deformation failure (buckling). To confirm that a structure will not deform to collapse the volume of vacuum, one must confirm that the potential energy increase associated with deforming the chamber (the deformation energy) is greater that the potential energy decrease associated with the corresponding reduction in the volume of vacuum (the vacuum energy). Those skilled in the art will know that this is an “energy balance” approach to stability analysis.
Either by theoretical analysis or experiment, it can be shown that the mode of deformation having the largest vacuum energy to deformation energy ratio is the “n=2” or “circle to ellipse” mode of deformation. FIG. 15 graphically depicts this mode. The circular section (dashed) is the undeformed state, while the elliptical section is the deformed state. The deformation parameter d gives the displacement of the unit-radius cylinder wall along the major and minor axes of the elliptical deformation.
The vacuum energy in this deformation mode can be calculated as:
      P    ⁢                  ⁢    Δ    ⁢                  ⁢    V    =            -              3        2              ⁢          d      2        ⁢    π    ⁢                  ⁢    P  
The deformation energy can be calculated as:
      +          3      8        ⁢      t    3    ⁢      d    2    ⁢  E  ⁢          ⁢  π
In this expression, E is the material's modulus of elasticity. In order that the sum of the deformation and vacuum energies be greater than zero (noting the sign change in the two expressions above), the following must be true:
  t  >            [                        4          ⁢          P                E            ]              1      3      
The required thicknesses can then be computed. These are presented in the table below:
TABLE IIIModulus ofMin. Wall Thickness (mm),MaterialElasticity (ksi)for a 1 m radius chamber2014-T6 Aluminum10,60017.7304 Stainless Steel28,00012.8Am 1004-T61 Magnesium6,48020.9Ti-6Al-4V Titanium17,40015.0Kevlar 49 FRP12,66616.7Carbon fiber composite16,66615.2
Comparing TABLE III to TABLE II, the reader will note that the wall thickness required for buckling stability is far greater than the maximum wall thickness allowed for positive buoyancy. Thus, one can conclude that it is not possible to construct a positively buoyant vacuum chamber using a conventional thin-walled cylinder (at least using any materials presently known).
In order for the thin-walled cylinder to achieve positive buoyancy, one must somehow bring together the minimum thickness required for stability and the maximum thickness allowed for weight concerns. Assuming a standard temperature (15 degrees Celsius) and specifying that the two requirements be compatible, one can write the following equation:
            (                        4          ⁢          P                E            )              1      3        ≤            1      2        ⁢                  ρ        air            ρ      
The air density is of course related to the pressure. One actually discovers that at a certain pressure, the equation can be satisfied. The following table provides this critical “buoyancy pressure” for several materials:
TABLE IVMaterialBuoyancy Pressure (atm)Ti-6Al-4V Titanium1131Kevlar 49 FRP248Carbon fiber composite255
Thus, the reader will observe that buoyancy is at least theoretically possible under very high pressure. Of course, it is much more desirable to achieve buoyancy at or below a pressure of 1 atmosphere. Obviously, the thin-walled cylinder approach cannot achieve this goal.
One can see in the equation above that if the modulus of elasticity can be increased, the pressure required for buoyancy would decrease. Of course, one cannot simply alter the modulus, as this is fixed for a given material. One can, however, alter the wall structure to produce the same result. For example, adding corrugations to the wall surface can increase the effective modulus by a factor of about nine and therefore decrease the critical pressure by a factor of about three.
One must be careful to preserve the strength in the circumferential direction. Because a cylinder under external pressure has twice as much compressive stress around its circumference compared to its long axis, it is acceptable (if using an isotropic building material) to lose 50% of the strength in the axial direction through changes in geometry (like corrugation) but it is undesirable to lose any strength in the circumferential direction. Thus, the corrugations should run around the circumference of the cylinder and not along its axis. For anisotropic materials, this technique is less effective because the fibers of the matrix are originally placed to provide strength only in the direction it is needed and thus any alteration of geometry that reduces strength in any direction (for an optimized fiber orientation) immediately requires weight increase.
Corrugation helps, but it cannot possibly achieve buoyancy at atmospheric pressure. Even the use of a complex “corrugation upon corrugation” approach—where the corrugated profile is itself corrugated—cannot achieve buoyancy. Clearly, a different approach is needed.
One can add internal pre-tensioned members for anti-buckling stabilization (in much the same way that tensioned bicycle spokes keep a bicycle wheel round). This technique is most effective against the “n=2” mode of deformation and thus, once it is employed, the most active mode of deformation may be the “n=3” mode (circle to pear shape) or some other mode. Unfortunately, the inclusion of such internal pre-tensioned members immediately adds structural weight. A relatively brief analysis indicates that it may be possible, through the use of multiple corrugations and internal stabilizers, to create a chamber which would achieve positive buoyancy at standard-air conditions. However, structural details would be very small and manufacture would be nearly impossible.
If, on the other hand, a design is created which places the walls primarily under tensile forces, most of the stability concerns are eliminated. Thus, a vacuum chamber wall design which experiences primarily tensile forces is desirable.